I am trying to help a student analyze several repeated-measures
designs and want to confirm that I am giving her accurate advice
concerning how to do the conservative F test when sphericity cannot be
assumed. She is using software (JMP) which apparently doesn't
estimate epsilon for the Huynh-Feld or Greenhouse-Geiser methods, and
she doesn't want to use the MANOVA approach (doesn't want to have to
explain it to her committee, I think).

Her basic design has one within-subjects factor (time) with 10 levels,
5 subjects, and no among-subjects factors. The unadjusted dfs would
be time=9, subject=4, time x subject (= residual) = 36. The "worst-
case" adjustment would compare the time F statistic (MS[T] / MS[TxS])
to the F distribution with 1 and 4 df, rather than 9 and 36.

From this starting point, there are 4 issues I am unsure about:

(1) The student also wants to test the subject effect, and I can't
find any discussion of whether or how to adjust the df for this test.
My intuition tells me we still would reduce the interaction (=
residual) df to 4 but wouldn't do anything to the subject df, so the
test would have 4 and 4 df. Is this correct?

(2) She wants to do multiple post-hoc comparisons among the times.
The only thing I could suggest for this, without assuming sphericity,
is to do all the pairwise combinations of paired t tests with
Bonferroni adjustment. Is there anything better?

(3) In one experiment she also has a second within-subjects treatment,
crossed with the within-subjects time factor. Happily this factor
only has 2 levels, so I think no new problems with nonsphericity
arise. But she can now separate subject x time and subject x
treatment terms from the residual (which would be equivalent to the
subject x time x treatment effect). I think her ANOVA table would be

source df F denom worst-case F dfs
Subjects 4 ?? ??
Time 9 S x T 1, 4
tReatment 1 R x T 1, 4
S x T 36 error 4, 4
S x R 4 error 4, 4
T x R 9 error 1, 4
"error" 36

I presume the denominator for the subjects F test would be a
composite, something like
MS[SxT] + MS[SxR] - MS[error]. But my main concern is the adjusted df
for the time, treatment, and time x treatment tests: am I correct that
all would be 1, 4?

(4) In one experiment there are multiple (3) measurements made on each
subject at each time point (this experiment has only the time and
subject factors, no other treatment). She can simply average these
subsamples and do all the analyses as above. But she also wants to
compare the among-measurement variance component to the among-subjects
variance component (and I suppose to the subject x time variance
component). Will variance component estimation be affected by
nonsphericity? Is so, what can be done to correct for it?

Thanks in advance for any advice.

Andy Taylor
Department of Zoology
University of Hawaii
tay... at (no spam) hawaii.edu